As in [1]}, [2]}, [3]}, we exploit that, due to the inextensibility constraint, only the normal components of the hydrodynamic force along the filament actually contribute to the fiber motion. In particular, using the parameterization (REF ), we may rewrite the first three equations of (REF ) as a closed system:
\(\dot{\mathbf {X}}_0 +\int _0^s \dot{\mathbf {e}}_{\rm t}(s^{\prime })\,ds^{\prime } &= -(1+\mu )({\bf I}+\gamma \mathbf {e}_{\rm t}\mathbf {e}_{\rm t}^{\rm T})\mathbf {h}(s)  \\\big ({\bf I}-\mathbf {e}_{\rm t}(s)\mathbf {e}_{\rm t}(s)^{\rm T}\big )\int _0^s\mathbf {h}(s^{\prime })ds^{\prime } &= \big ({\bf I}-\mathbf {e}_{\rm t}(s)\mathbf {e}_{\rm t}(s)^{\rm T}\big )\bigg (\mathbf {X}_{sss}-(\kappa _0)_s\mathbf {e}_{\rm n}-\frac{\mu }{1+\mu }\xi _s\mathbf {e}_{\rm n}\bigg )  \\\delta \dot{\xi }&= -\xi + \kappa - \kappa _0\,.\) 
